Does pollen have an allergenic "expiry date"?

It's spring here in New Zealand, and all our heatpumps in the house are full of pollen. Whenever we switch them on, the sneezing and teary-eyed thing begins. We've tried cleaning the filters, but to no avail. Can we expect that pollen in the heatpumps to eventually "die" and stop making us sneeze, or will it keep doing that for years to come?